In this brief, the optimally pruned extreme learning machine (OP-ELM) methodology is presented. It is based on the original extreme learning machine (ELM) algorithm with additional steps to make it more robust and generic. The whole methodology is presented in detail and then applied to several regression and classification problems. Results for both computational time and accuracy (mean square error) are compared to the original ELM and to three other widely used methodologies: multilayer perceptron (MLP), support vector machine (SVM), and Gaussian process (GP). As the experiments for both regression and classification illustrate, the proposed OP-ELM methodology performs several orders of magnitude faster than the other algorithms used in this brief, except the original ELM. Despite the simplicity and fast performance, the OP-ELM is still able to maintain an accuracy that is comparable to the performance of the SVM. A toolbox for the OP-ELM is publicly available online.